AP Precalculus: Continuity
Master the definition of continuity, types of discontinuity, and interval analysis
📚 Understanding Continuity
A function is continuous when you can draw its graph without lifting your pencil. Formally, continuity requires three conditions to be met at a point. Understanding continuity helps identify where functions behave predictably and where special behavior (holes, jumps, asymptotes) occurs.
1 Formal Definition of Continuity
A function f(x) is continuous at x = a if and only if all three of the following conditions are satisfied:
(defined at a)
(limit exists)
(limit equals value)
Function: \(f(x) = \frac{x^2 - 4}{x - 2}\) at x = 2
Condition 1: f(2) = \(\frac{0}{0}\) — undefined ✗
Conclusion: Not continuous at x = 2 (fails first condition)
If ANY of the three conditions fails, the function is discontinuous at that point. Always check all conditions when analyzing continuity.
2 Types of Discontinuity
When a function is not continuous at a point, the type of discontinuity tells us exactly how continuity fails.
Removable vs Non-Removable
Removable: Can be "fixed" by redefining f(a). Jump/Infinite: Cannot be fixed — discontinuity is essential.
How to Identify
Factor rational functions. Common factors in numerator/denominator create removable discontinuities (holes).
Removable: \(f(x) = \frac{x^2-1}{x-1}\) at x = 1 (hole at (1, 2))
Jump: \(f(x) = \begin{cases} x & x < 0 \\ x + 2 & x \geq 0 \end{cases}\) at x=0
Infinite: \(f(x) = \frac{1}{x-3}\) at x = 3 (vertical asymptote)
3 One-Sided Continuity
One-sided continuity means the function is continuous when approaching from only one direction — useful at endpoints of intervals.
A function is continuous at x = a if and only if it is BOTH left-continuous AND right-continuous at a.
Function: \(f(x) = \sqrt{x}\)
At x = 0: Right-continuous (approaching from positive values)
Not left-continuous because \(\sqrt{x}\) is undefined for x < 0
4 Continuity on Intervals
A function is continuous on an interval if it is continuous at every point in that interval, with appropriate one-sided continuity at endpoints.
Open Interval (a, b)
Continuous at every point x where a < x < b
Closed Interval [a, b]
Continuous on (a, b), plus right-continuous at a, left-continuous at b
Functions Continuous on Their Domains
- Polynomials: Continuous everywhere \((-\infty, \infty)\)
- Rational functions: Continuous except where denominator = 0
- Trigonometric: sin(x), cos(x) continuous everywhere; tan(x) has asymptotes
- Exponential: \(e^x\), \(a^x\) continuous everywhere
- Logarithmic: Continuous on \((0, \infty)\)
5 Analyzing Points of Discontinuity
To identify and classify discontinuities, examine the function systematically at suspicious points.
Where to Look for Discontinuities
- Rational functions: Where denominator equals zero
- Piecewise functions: At the boundary points between pieces
- Radical functions: Where radicand equals zero (for even roots)
- Logarithmic functions: Where argument equals zero
Step-by-Step Analysis
(zeros of denom, boundaries)
(defined, limit exists, equal)
(removable, jump, infinite)
Function: \(f(x) = \begin{cases} x^2 & x < 1 \\ 2x - 1 & x \geq 1 \end{cases}\)
Check x = 1:
• \(f(1) = 2(1) - 1 = 1\) ✓ (defined)
• Left limit: \(\lim_{x \to 1^-} x^2 = 1\)
• Right limit: \(\lim_{x \to 1^+} (2x-1) = 1\)
• Limits equal, so \(\lim_{x \to 1} f(x) = 1 = f(1)\) ✓
Conclusion: Continuous at x = 1 ✓
📋 Quick Reference
Continuity at a
\(\lim_{x \to a} f(x) = f(a)\)
Right-Continuous
\(\lim_{x \to a^+} f(x) = f(a)\)
Left-Continuous
\(\lim_{x \to a^-} f(x) = f(a)\)
Removable
Limit exists but ≠ f(a)
Jump
Left limit ≠ Right limit
Infinite
Limit = ±∞